Download Fuzzy pgprw-continuous maps and fuzzy pgprw

International Journal of Statistics and Applied Mathematics 2016; 1(1): 01-07
ISSN: 2456-1452
Maths 2016; 1(1): 01-07
© 2016 Stats & Maths
www.mathsjournal.com
Received: 01-03-2016
Accepted: 02-04-2016
RS Wali
Department of Mathematics,
Bhandari and Rathi College
Guledagudd-587203, Karnataka,
India.
Vivekananda Dembre
Department of Mathematics,
Rani Channamma University,
Belagavi-591156, Karnataka,
India.
Fuzzy pgprw-continuous maps and fuzzy pgprwirresolute in fuzzy topological spaces
RS Wali, Vivekananda Dembre
Abstract
In this paper, we introduce the concept of fuzzy pgprw-continuous maps and fuzzy pgprw-irresolute
maps in fts. We prove that the composition of two fuzzy pgprw-continuous maps need not be fuzzy
pgprw-continuous and study some of their properties.
Keywords: Fuzzy pgprw-continuous maps, fuzzy pgprw-irresolute maps.
Introduction
The concept of a fuzzy subset was introduced and studied by L.A. Zadeh in the year 1965. The
subsequent research activities in this area and related areas have found applications in many
branches of science and engineering. In the year 1965, L.A.Zadeh [1]. introduced the concept of
fuzzy subset as a generalization of that of an ordinary subset. The introduction of fuzzy subsets
paved the way for rapid research work in many areas of mathematical science. In the year
1968, C.L.Chang [2] introduced the concept of fuzzy topological spaces as an application of
fuzzy sets to topological spaces . Subsequently several researchers contributed to the
development of the theory and applications of fuzzy topology. The theory of fuzzy topological
spaces can be regarded as a generalization theory of topological spaces. An ordinary subset A
or a set X can be characterized by a function called characteristic function
A
:X
[0,1] of A, defined by
A
(x) = 1, if x
A.
= 0, if x A.
Thus an element x
in A if
A
(x) = 1 and is not in A if
A
(x) = 0. In general if X is a
set and A is a subset of X then A has the following representation. A = { ( x ,
A
(x)): x X},
here A (x) may be regarded as the degree of belongingness of x to A, which is either 0 or 1.
Hence A is the class of objects with degree of belongingness either 0 or 1 only. Prof.
L.A.Zadeh [1] introduced a class of objects with continuous grades of belongingness ranging
between 0 and 1 ; he called such a class as fuzzy subset. A fuzzy subset A in X is characterized
as a membership function
[0,1] , which associates with each point in x a real number A(x) between 0
A:X
and 1 which represents the degree or grade membership of belongingness of x to A.
The purpose of this paper is to introduce a new class of fuzzy sets called fuzzy pgprw-closed
sets in fuzzy topological spaces and investigate certain basic properties of these fuzzy sets.
Among many other results it is observed that every fuzzy closed set is fuzzy pgprw-closed but
not conversely. Also we introduce fuzzy pgprw-open sets in fuzzy topological spaces and
study some of their properties.
Correspondence:
RS Wali
Department of Mathematics,
Bhandari and Rathi College
Guledagudd-587203, Karnataka,
India.
1. Preliminaries
1.1Definition: :[1] A fuzzy subset A in a set X is a function A : X → [0, 1]. A fuzzy subset in
X is empty iff its membership function is identically 0 on X and is denoted by 0 or ϕ. The set
X can be considered as a fuzzy subset of X whose membership function is identically
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1 on X and is denoted by x or Ix. In fact every subset of X is a fuzzy subset of X but not conversely. Hence the concept of a fuzzy
subset is a generalization of the concept of a subset.
1.2 Defnition: [1] If A and B are any two fuzzy subsets of a set X , then A is said to be included in B or A is contained in B iff
A(x) ≤ B(x) for all x in X. Equivalently, A ≤ B iff A(x) ≤ B(x) for all x in X.
1.3 Defnition: [1] Two fuzzy subsets A and B are said to be equal if A(x) = B(x) for every x in X.
Equivalently A = B if A(x) = B(x) for every x in X.
1.4 Defnition: [1] The complement of a fuzzy subset A in a set X, denoted by A′ or 1 − A, is the fuzzy subset of X defined by
A′(x) = 1 − A(x) for all x in X. Note that (A′)′ = A.
1.5 Defnition: [1] The union of two fuzzy subsets A and B in X, denoted by A ∨ B, is a fuzzy subset in X defined by (A ∨ B)(x) =
Max{ A(x), B(X)} for all x in X.
1.6 Defnition: [1] The intersection of two fuzzy subsets A and B in X, denoted by A ∧ B, is a fuzzy subset in X defined by (A ∧
B)(x) = Min{A(x), B(x)} for all x in X.
1.7 Defnition: [1] A fuzzy set on X is ‘Crisp‘ if it takes only the values 0 and 1 on X.
1.8 Defnition: [2] Let X be a set and be a family of fuzzy subsets of (X,
following conditions.
(i) ϕ ; X ∈ : That is 0 and 1 ∈
(ii) If Gi ∈
is called a fuzzy topology on X iff
satisfies the
for i ∈ I then ∨ Gi ∈
i
(iii) If G,H ∈
then G ∧ H ∈
The pair (X, ) is called a fuzzy topological space (abbreviated as fts). The members of
set A in X is said to be closed iff 1 − A is an fuzzy open set in X.
are called fuzzy open sets and a fuzzy
1.9 Remark: [2] Every topological space is a fuzzy topological space but not conversely.
1.10 Defnition: [2] Let X be a fts and A be a fuzzy subset in X. Then ∧ {B : B is a closed fuzzy set in Xand B ≥ A} is called the
closure of A and is denoted by A or cl(A).
1.11 Defnition: [2]Let A and B be two fuzzy sets in a fuzzy topological space (X, )and let A ≥ B. Then B is called an interior
fuzzy set of A if there exists G ∈ such that A ≥ G ≥ B, the least upper bound of all interior fuzzy sets of A is called the interior of
A and is denoted by A0.
1.12 Definition: [3] A fuzzy set A in a fts X is said to be fuzzy semiopen if and only if there exists a fuzzy open set V in X such
that V A cl(V).
1.13 Definition: [3] A fuzzy set A in a fts X is said to be fuzzy semi-closed if and only if there exists a fuzzy closed set V in X
such that int(V) A V. It is seen that a fuzzy set A is fuzzy semiopen if and only if 1-A is a fuzzy semi-closed.
1.14 Theorem: [3] The following are equivalent:
(a)
is a fuzzy semiclosed set,
(b)
C
is a fuzzy semiopen set,
(c) int(cl( ))
.
(b) int(cl( ))
c
1.15 Theorem: [3] Any union of fuzzy semiopen sets is a fuzzy semiopen set and (b) any intersection of fuzzy semi closed sets is
a fuzzy semi closed.
1.16 Remark: [3]
(i) Every fuzzy open set is a fuzzy semiopen but not conversely.
(ii) Every fuzzy closed set is a fuzzy semi-closed set but not conversely.
(iii) The closure of a fuzzy open set is fuzzy semiopen set
(iv) The interior of a fuzzy closed set is fuzzy semi-closed set
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International Journal of Statistics and Applied Mathematics
1.17Definition: [3] A fuzzy set
of a fts X is called a fuzzy regular open set of X if int(cl( ).= .
1.18 Definition: [3] A fuzzy set
of fts X is called a fuzzy regular closed set of X if cl (int( )) = .
1.19 Theorem: [3] A fuzzy set
of a fts X is a fuzzy regular open if and only if
C
fuzzy regular closed set.
1.20 Remark: [3]
(i) Every fuzzy regular open set is a fuzzy open set but not conversely.
(ii) Every fuzzy regular closed set is a fuzzy closed set but not conversely.
1.21 Theorem: [3]
(i) The closure of a fuzzy open set is a fuzzy regularclosed.
(ii) The interior of a fuzzy closed set is a fuzzy regular open set.
1.22 Definition: [4] A fuzzy set
in X.
in fts X is called fuzzy rwclosed if cl ( )
1.23 Definition [5]: A fuzzy set
set in X.
in fts X is called fuzzy pgprw closed if p-cl ( )
1.24 Defintion [5]: A fuzzy set
whenever
of a fts X is fuzzy pgprw-open set, if it’s complement
and
whenever
c
is regular semi-open
and
is rg
open
is a fuzzy pgprw-closed in fts X.
1.25 Definition [2]: Let X and Y be fts. A map f: X Y is said to be a fuzzy continuous mapping if f -1( ) is fuzzy open in X for
each fuzzy open set
in Y.
Y is said to be a fuzzy almost continuous mapping if f -1( ) is fuzzy
1.26 Definition [6]: Let X and Y be fts. A map f: X
open in X for each fuzzy regular open set
in Y.
Y is said to be a fuzzy irresolute if f -1( ) is fuzzy semi-open set in
1.27 Definition [6]: Let X and Y be fts. A map f: X
X for each fuzzy semi-open set
in Y.
1.28 Definition [2]: Let X and Y be fts. A map f: X
f -1( ) is fuzzy semi-open set in X for each fuzzy open set
1.29 Definition [4]: Let X and Y be fts. A map f: X
f -1( ) is fuzzy rw-open set in X for each fuzzy open set
Y is said to be a fuzzy semi-continuous if
in Y.
Y is said to be a fuzzy rw-continuous if
in Y.
Y is said to be a fuzzy completely semi-continuous if f -1( ) is
1.30 Definition [7]: Let X and Y be fts. A map f: X
fuzzy regular semi-open set in X for each fuzzy open set
in Y.
2. Fuzzy pgprw-continuous maps and fuzzy pgprw-irresolute maps in fuzzy topological spaces
Definition 2.1: Let X and Y be fts. A map f: X
fuzzy open set in Y is fuzzy pgprw-open in X.
Theorem 2.2: If a map f: (X, T)
Proof: Let
Y is said to be fuzzy pgprw-continuous map if the inverse image of every
(Y, T) is fuzzy continuous, then f is fuzzy pgprw-continuous map.
be a fuzzy open set in a fts Y. Since f is fuzzy continuous map f-1( ) is a fuzzy open set in fts x,as every open set is
fuzzy pgprw-open ,we have f-1( ) is fuzzy a pgprw-open set in fts X. Therefore f is fuzzy continuous map.
The converse of the above theorem need not be true in general as seen from the following example
Example 2.3: Let X=Y= {a, b, c,d} and the functions
,
:X
(x) = 1 if x = a
0 otherwise
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[0, 1] be defined as
International Journal of Statistics and Applied Mathematics
(x) = 1 if x = b
0 therwise
= 1 if x = a,b
0
otherwise
(x) = 1 if x = a,b,c
0
Otherwise.
Consider T= {1, 0, ,
}, = {1,0, ,
}.Now (X,T) and (Y, ) are the fts. Define a map
f:(X,T)
(Y,T) by f(a)=c,f(b)=a,f(c)=b,f(d)=d, then f is fuzzy pgprw-continuous map but not fuzzy continuous map as the
inverse image of the fuzzy set in (Y, ) is :X
[01}defined as
(x) = 1 if x = b,c
0 otherwise.
This is not a fuzzy open set in (X, T).
Theorem 2.4: A map f: ( X,T)
(Y, ) is fuzzy pgprw-continuous map iff the inverse image of every fuzzy closed set
in a fts Y is a fuzzy pgprw closed set in fts X.
Proof: Let
be a fuzzy closed set in a fts Y then
c
is fuzzy open in fts Y. Since f is fuzzy pgprw-continuous map,f
pgprw-open in fts X but f -1 ( c) = 1- f -1 ( c) and so f – 1(
c
( c) is
is a fuzzy pgprw-closed set in fts X.
Conversely, Assume that the inverse image of every fuzzy closed set in Y is fuzzy pgprw closed in fts X. Let
set in fts Y then
-1
be a fuzzy open
is fuzzy closed in Y;
by hypothesis f – 1( c)= 1- f – 1(
pgprw-continuous map.
is fuzzy pgprw closed in X and so f – 1(
Theorem 2.5: If a function f: (X, T)
Proof: Let a function f: (X, T)
is a fuzzy pgprw-open set in fts X. Thus f is fuzzy
(Y, ) is fuzzy almost continuous map, then it is fuzzy pgprw-continuous map.
(Y, ) be a fuzzy almost continuous map and
– 1
fuzzy regular open set in fts X. Now, f (
.Therefore f is fuzzy pgprw-continuous map.
be fuzzy open set in fts Y. Then f – 1(
is a
is fuzzy pgprw-open in X, as every fuzzy regular open set is fuzzy pgprw-open
The converse of the above theorem need not be true in general as seen from the following example.
Example 2.6: consider the fts (X, T) and (Y, ) as defined in example 2.3. define a map
f: (X,T)
map.
(Y, ) by f(a)= c,f(b)=a,f(c)=b,f(d)=d. Then f is fuzzy pgprw-continuous map but it is not almost continuous
Example 2.7: fuzzy semi-continuous maps and fuzzy pgprw-continuous maps are independent as seen from the following
examples.
Example 2.8: Let X= Y={a, b, c,d} and the functions
,
:X
[0, 1] be defined as
(x) = 1 if x = a
0 otherwise
(x) = 1 if x = b
0 therwise
= 1 if x = a,b
0 otherwise
(x) = 1 if x = a, b, c
0 otherwise.
Consider T={1,0, ,
},
= {1,0,
.Let map f: X
Y defined by f(a)=b,f(b)=b,f(c)=a,f(d)=c, then f is fuzzy pgprw-
continuous map but it is not fuzzy semi continuous map as the inverse image of fuzzy set
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c
in (Y, ) is
International Journal of Statistics and Applied Mathematics
(x) = 1 if x = a, b, d
0 otherwise.
This is not a fuzzy semi closed set in fts X.
Example 2.9:
Let X= Y= {a, b, c} and the functions ,
:X
[0, 1] be defined as
(x) = 1 if x = a
0 otherwise
(x) = 1 if x = b, c
0 therwise
Consider T= {1,0, , } ,
= {1,0,
.Let map f: X
Y defined by f(a)=b,f(b)=a,f(c)=c.
c
then f is fuzzy semi-continuous map but f is not fuzzy pgprw continuous map as the inverse image of fuzzy set
in (Y, ) is
(x) = 1 if x = a,c
0 otherwise.
This is not fuzzy pgprw-closed set in fts X.
Remark 3.0: fuzzy rw-continuous maps and fuzzy pgprw-continuous maps are independent as seen from the following examples.
Example 3.1: Let X= {a, b, c,d},Y={a,b,c} and the functions
,
:X
[0, 1] be defined as
(x) = 1 if x = a
0 otherwise
(x) = 1 if x = b
0 therwise
= 1 if x = a, b
0 Otherwise.
(x) = 1 if x = a, b, c
0
otherwise
Consider T= {1,0, ,
}, = {1,0,
.Let map f:X
Y defined by f(a)=b, f(b)=a, f(c)=a, f(d)=c, then f is fuzzy pgprw-
continuous map but it is not fuzzy rw continuous map as the inverse image of fuzzy set
c
in (Y, ) is
(x) = 1 if x = a ,d
0 otherwise.
This is not fuzzy rw-closed set in fts X.
Example 3.2: Let X= Y={a, b, c} and the functions
,
:X
[0, 1] be defined as
(x) = 1 if x = a
0 otherwise
(x) = 1 if x = b, c
0 therwise
Consider T= {1,0, , } ,
= {1,0,
.Let map f: X
Y defined by f(a)=b, f(b)=a, f(c)=c.
Then f is fuzzy rw-continuous map but f is not fuzzy pgprw continuous map as the inverse image of fuzzy set
(x) = 1 if x = a,c
0 otherwise.
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c
in (Y, ) is
International Journal of Statistics and Applied Mathematics
This is not fuzzy pgprw-closed set in fts X.
Remark 3.3: From the above discussion and known results we have the following implication
Theorem 3.4: If a function f : (X,T)
is fuzzy continuous map.
Proof: Let a function f : (X,T)
(Y, ) is fuzzy pgprw-continuous map and fuzzy completely continuous map then it
(Y, ) be a fuzzy pgprw-continuous map and fuzzy completely semi-continuous map. Let
be a fuzzy closed set in fts Y. Then f – 1( ) is both fuzzy closed set in fts Y.Then f – 1( ) is both fuzzy rg -open and fuzzy pgprwclosed set in fts X .If a fuzzy set of fts X is both fuzzy rg and fuzzy pgprw-closed then
is a fuzzy closed set in fts X .Therefore f is fuzzy continuous map.
Theorem 3.5: If f: (X,T)
(Y, ) is fuzzy pgprw-continuous map and g: (Y, )
then their composition gof:(X, T)
Proof: Let
is a fuzzy closed in fts X thus f – 1( )
(Z, ) is fuzzy continuous map,
(Z, ) is fuzzy pgprw-continuous map.
be a fuzzy open set in fts Z. Since g is fuzzy continuous map, g -1( ) is fuzzy open set in fts Y. Since f is fuzzy
pgprw-continuous map,f – 1( g -1( )) is a fuzzy pgprw-open set in fts X.
But (gof)- 1( )= f – 1( g -1( )) thus gof is fuzzy pgprw-continuous map.
Definition 3.6: Let X and Y be fts. A map f: X
Y is said to be a fuzzy pgprw-irresoulute map if the inverse image of every
fuzzy pgprw-open in Y is a fuzzy pgprw-open set in X.
Theorem 3.7: If a map f: X
Proof: Let
Y is fuzzy pgprw- irresolute map then it is fuzzy pgprw-continuous map.
be a fuzzy open set in Y. since every fuzzy open Set is fuzzy pgprw-open,
Fuzzy pgprw-open set in Y. Since f is fuzzy pgprw- irresolute map,f
continuous map.
is a
– 1
( ) is fuzzy pgprw-open in x. Thus f is fuzzy pgprw-
The converse of the above theorem need not be true in general as seen from the following example.
Example 3.8:Let X={a, b, c,d} , Y={a,b,c} the functions
,
:X
[0, 1] be defined as
(x) = 1 if x = a
0 otherwise
(x) = 1 if x = b
0 therwise
= 1 if x = a, b
0 otherwise
(x) = 1 if x = a, b, c
0 otherwise
Consider T= {1,0, ,
},
= {1,0,
.Let map f: X
Y defined by f(a)=b, f(b)=a, f(c)=a,f(d)=c, then f is fuzzy pgprw-
continuous map but it is not fuzzy pgprw-irresolute map.Since for the fuzzy pgprw-closed set : Y
(x) = 1 if x = b
0 otherwise in Y
f – 1( ) = is not fuzzy pgprw-closed in (X, T).
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[0,1] defined by
International Journal of Statistics and Applied Mathematics
Theorem 3.9: Let X, Y, Z be fts. If f: X
Y is fuzzy pgprw-irresolute map and g: Y
Z
is fuzzy pgprw-continuous map then their composition gof: X
Z is fuzzy pgprw-continuos map.
Proof: Let
be any fuzzy open set in fts Z, Since g is fuzzy pgprw-continuous map, ( g -1( )) is a fuzzy pgprw-irresolute map f –
1
((g -1( )) is a fuzzy pgprw-open set in fts X
But (gof)- 1( ) = f – 1( g – 1( )).Thus gof is fuzzy pgprw-continuous map.
Theorem 3.10: Let X, Y, Z be fts and f: X Y and g: Y
Z be fuzzy pgprw-irresolute maps then their composition maps
then their composition gof: X
Z is fuzzy pgprw-irresoulte map.
Proof: Let
be a fuzzy pgprw-open set in fts Z. since g is fuzzy pgprw-irresolute map, g -1( )
is a fuzzy pgprw-open set in fts Y. since f is fuzzy pgprw-irresolute map, f – 1( g – 1( )) is a fuzzy
Pgprw-open set in fts X But (gof) – 1( ) = f – 1 (g – 1( )). Thus gof is fuzzy pgprw-irresolute map.
Conclusion : In this paper, a new class of maps called Fuzzy Pgprw-Continuous maps and Fuzzy pgprw-irresolute in fuzzy
topological spaces are introduced and investigated.In future the same process will be analyzed for Fuzzy pgprw-properties.
References
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4. Wali RS. On some topics in general and fuzzy topological spaces; ph. d thesis, Karnatak university Dharwad, 2007.
5. Wali RS, Vivekananda Dembre. Fuzzy pgprw-closed sets and Fuzzy pgprw-open sets in Topological Spaces; journal of
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6. Azad kk. On fuzzy semi-continuity, fuzzy almost continuity and fuzzy weak continuity, j. math. anal app 1981; 829:14-81.
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